Quantum mechanical bootstrap without inequalities: SYK bilinear spectrum

TL;DR

This paper introduces a direct bootstrap method for determining the spectrum of a quantum system related to the SYK model, avoiding positivity constraints by using fractional powers of operators.

Contribution

It develops a novel direct bootstrap approach that accurately computes the spectrum without relying on positivity, applicable to SYK bilinear operators.

Findings

Roots converge to exact eigenvalues with higher truncation

Method distinguishes SYK spectrum without positivity constraints

Provides a new way to analyze quantum spectra

Abstract

We study a quantum mechanical system whose spectrum coincides with that of bilinear operators of the Sachdev-Ye-Kitaev model. The standard positivity-based quantum mechanical bootstrap is degenerate with respect to the boundary data: it does not distinguish the boundary conditions that select the SYK spectrum, and hence is insufficient to determine the eigenvalues. Instead, by considering fractional powers of operators, we obtain constraint equations that determine the spectrum without imposing positivity. The resulting roots converge to exact eigenvalues as the truncation order increases. We call this the direct bootstrap.